Multidimensional Simulations of Incompressible Turbulent Flows Using a Godunov-type Scheme for Reynolds-averaged Navier–Stokes Second-Order Moment Equations

Turbulence modelling for incompressible flows remains challenging when strong anisotropy and intercomponent energy transfer are essential, as in thermal-hydraulics or atmospheric boundary layers. To address the limitations of eddy-viscosity closures, we consider the second-order turbulence-moment equations that transport the full Reynolds-stress tensor, with intercomponent energy redistribution modelled by the classical Rotta return-to-isotropy closure. A key difficulty in numerical simulations of such equations lies in maintaining the realisability of the Reynolds-stress tensor–symmetry and positive semi-definiteness–at the discrete level to avoid loss of hyperbolicity and numerical breakdown. We introduce a global two-step algorithm for multidimensional incompressible Reynolds-averaged Navier–Stokes equations: (i) an explicit Godunov-type convective predictor for stable transport of Reynolds stresses; and (ii) an implicit correction step applied to the mean velocity to enforce incompressibility and account for pressure, while the Reynolds stresses are updated using a dedicated diffusion-source operator. A key innovation is a semi-implicit, realisability-preserving strategy embedded in both the convection step and the integration of the Rotta source term, guaranteeing positive semi-definiteness in every cell and substep. Validation includes tests of the Rotta model using ordinary differential equations, where the Reynolds stress trajectories remain within the Lumley triangle, and full simulations of a plane mixing layer. Results capture turbulence onset, self-similar growth, and Reynolds stress evolution, with good agreement to experimental data even on coarse meshes.